IMC 2013 · Problem 2

Day 120th IMC · Blagoevgrad, Bulgaria

Statement

Let $f : \mathbb{R} \to \mathbb{R}$ be a twice differentiable function. Suppose $f(0) = 0$ . Prove that there exists $\xi \in (-\pi/2, \pi/2)$ such that

f''(\xi) = f(\xi)(1 + 2\tan^2 \xi).

Official solution

Hidden so you can work on the problem first.

Proposed by Karen Keryan, Yerevan State University, Yerevan, Armenia.